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Unformatted text preview: reserves direct at the General, Health, Law, Oxford, and Theology Libraries The United States copyright law (Title 17 of the US Code) governs the making of
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reserve items are copyrighted Math 111
Review Questions for Test 2 1, Sketch the graph of each of the following functions. Label all important points and featuresl xx . . .d . b x3
omit ooncawt 001131 emtxons =5
x24 ( y ) () y 20;“) (2!) ya (0) y a: 3x5  5x3 + 3 (omit determining X~intercepts) (ye—2? . . . . 4 3
(d) y a (omit concav1t oonmderations) (e) y =2 x + 4x
T36 Y
(0 y w 2x3  3x2 12x +18 (3) )1 == x%(x + 5)}5 (omit concavity considerations)
(h) x2 (0 x550: + 4)
y m y
. 3 . . . . 2x2 + 5
(3) y = x ~12x + 2 (onnt detennimng X4ntercepts) (k) y = x2 +3
~x
(I) y=x5<x+4) (m))’=“*““~§
(x ~ 1)
)6 2', i . . . 5 3
(n) y = x (x + 1) (omit concavuy consxderations) (o) y =2 3x — 5x (p) y = x36 (12 ~ JOE (omit concavity considerations) II. Evaluate the following limits (in case of an inﬁnite limit, deteimine the Sign and write
either +00 or ~ 00 , as approppriate): 2 3
. x —4 . x +8 . Inc71
l «y—wv b lm 0 km
(a) x1512 2: +4 () xiv2 x+2 () x~>7" x—7
. 3— x+6 . x . x+1
d l W e l m hm ~§—
()xlﬂls x—3 () Pllﬂ+ x+1 (0 x~v3+ x —9
2 3
x v2x+1 1 3«2x
1 h 1' ~ i lim
(g) x1311“? 3~4x2 x333“ x x~—>00 x2+1
. . x . . 2
1m §—— k 11m c053 1 km sec x
O) x—iw: x +1 () x6311 x 0 x»)?
x%
(m) lim tanx (n) lim cscx (0) lim — x435.“ x~w2x+ w+°o x+1 x2+X+12 . ;+1 e I2‘xl
' W— 1
(p) :3 3x2—21x+36 (q) x193) x3+125 (r) x1212 x+2
X
. . x3 —3x . ex +3
(8) x3131§+ tanzx * (t) x2312" X+2 (U) x1210 2ex—1
. 3x2 + sinx
(V) £1300 4x3—8
Answers:
(a) 0 (b) 12 (c) +1 (d) —61 (e) «00 (0 +00
0 (g) ~21: (h) ~00 (0 +00 (3)0 (k) «1
an (m) —00 (n) +00 (o) 0 (p) % (q) 215
(r) 0 (s) —00 to ~00 (u) 5 (v) 0 Ill. Rate "Problems 1. Air is being pumped into a spherical balloon at the rate of
81: in3l min. Find the rate of change of the radius when the surface area is 161: in2. (1/2 inlmin) , 2. Water is being poured into a conical container (vertex down)
which is 12 feet high and which has a diameter (at the top) of 8 feet.
If the water level is rising at the rate of 1/2 ftlsec, find the rate at
which water is being poured into the tank at the instant that the water in the container is 6 feet deep. (V = 1/31rr2h) (27r ft3lsec) 3. A pebble is dropped into a calm pond, causing ripples in the
form of concentric circles. The ratius r of the outer circle ripple is
increasing at a constant rate of 1 foot per second. When the radius
is 4 feet, at what rate is the total area A of the disturbed water increasing? (8:: ftZIsec) 4. Water runs into a conical tank at the rate of 9 ft3lmin. The tank stands vertex down and has a height of 10 ft and base radius of
5 feet. How fast is the water level rising when the water is 6 feet deep? (v = 1/37rr2h) (1/7: ftlmin) 5. At 9:00 am. car A is. headed north at 40 miles per hour and
is located 120 miles due south of car B, which is heading east at 15
miles per hour. How fast is the distance between the cars changing at 11:00 a.m.? (23 mph) '6. A man 6 feet tall walks away from a street light 15 feet
high at the rate of 6 ftlsec. How fast is the far end of his shadow moving when he is 30 feet from the light pole? (10 ftlsec) 7. Water is flowing out of a conical tank (vertex down) of
{height 10 feet and radius 6 feet in such a waythat the water level
is falling 1/2 foot per minute. How fast is the volume of water in
the tank decreasing when the wter in the tank is 5 feet deep? (91:12 ft3lmin) 8. A man standing on a pier is pulling in his boat by means of a
rope attached to the boat's how. If the pier is. 5 feet above the
water and the rope is being pulled in at 2 ftlsec, how fast is the
boat approaching the front of the pier when 13 feet of rope are still
out? (1316 ftlsec) ' 9. Two truck convoys leave a depot. Convoy A travels east at
40 miles per hour and convoy B travels north at 30 miles per hour.
How fast is the distance between the convoys changing 6 min later,
when convoy Ale 4 miles fromthe depot and convoy B is 3 miles '
from the depot? (50 miles per hour) 10. Sand is being poured into a conical shape at the rate of 6
ft3lmin. The diameter of the base is always 3 times the altitude.
How fast will the circumference of the base be changing when the
pile is 5 feet high? (8/25 ftlmin) IV. Determine (and explain) whether or not the Mean Value Theorem is applicable to the given
function over the given interval. If the theorem is applicable, ﬁnd the number c guaranteed by the theorem.
(a) fem/Xi" Over [1,5]
(b) foo: x24x over [2,6]
to f(x)=« x734 over [4,1]
(d) feast/2:? over [6,51 (e) f(x)=8vx3 over [—2,1] Aw (a) f (x) is continuous over [1,5] and f ’(x) = 2 is defined over (1,5); c = 2 (b) f (x) is continuous over [2,6] and f ’(x) = 2x — 4 is defined over (2,6); c = 4 (c) The Mean Value Theorem does not apply because f ’(x) = 5—32 is not defined at x = O.  x
(d) f (x) is continuous over [6,5] and f ' (x) = “rm” is defined over (6,5); 0 = 0
25  x2 (e) f (x) is continuous over [—2,1] and f '(x) = ~3x2 is defined over (~2,l); c = —1 V. Differentiation (a) Given :52 ~ x2 = y2 + cosxy , find Si)— in terms of x and y, and ﬁnd the equation of the normal line to the graph of this muation at the point (1, 3:). £12: __ ysinxy v2x ; Normalline: «ar==:rx1
dx 2y—xsinxy y ( ) Ans. d
(b) Given ln(1* xy2)== y+3, find a? in terms of x and y. dy y2 E=xy2—2xy~1 (0) Given xy + y3 a 5, find 32' and y" in terms of X and y. . d .
(d) Given xsjn2y=ycoszx, ﬁnd 2:: 111 terms of x and y. dy sin2 y + 232005 xsin x
dx cos x— ZXSHlyCOSy (e) Given y = (2— x)%(x + 2)%, find iiiXX in simplest terms. %
ARE. ﬂ=«(7x+6)(2~—x)
dx 3(x+2)z d
(0 Given 3 x3y53tanx~coty+mfindﬁintennsofxand y. _ x . ,, . .
(g) Gwen f(x)=‘—/=;§=:T,f1nd f (x) m Simplest terms ,, ~3x
Alﬁ f (x) =W Math 111  Test 2  Fall 1998 Honor Pledge K E Y Show all work . Write legibly and iucidiy. Totai Points: 110 1. For each of the functions given below, you are to sketch the graph of the function. Label all
important points and features of the graph Provide the additional information requested for each graph, writing NONE when this is appropriate. Show all work on separate paper
[25 points each] a :2:
() y 8*x3 Asymptote (s): x 1" 1L ' : ~ 1
Point (3) where the tangent line is horizontal: ( 02 0)
Relative maxima; m Relative minima: m ;
Interval (s) where the graph is concave downward: "' J; 0 9 0°) I“ .2435”
g (8~x’)"
we,»
++++ I++~H~ 1b+++ ;y
o 2.
in VA
% " : 7693(28154'
(9’43)?
Wat
+ +++— + ~.~~ —— — {t F98111T2r2 (b) y a: x % (2x — 3)?’g [Omit concavit considertions]
y Dwain; QM— MAL) Interval (s) ofinerease: C“ 00, 0) f 01 12:) g 2 0°) Interval (s) of decrease: J' x 3/2. . . .L 3
Pomt (s) where the tangent line IS horizontal: 2. 7 (3":
0 Point (8) where the tangent lin vertical: 0 O 1‘ eis
. . . é, '
Relatlve mlmma: ( 229.} V . . I 3 ‘
Relauve mamma: ﬂ “5 2 i“ 2 F98111T2«3 2. Helium is leaking from a spherical balloon [V x §3r31 . The balloon's surface area [A = 4mg] is shrinking at the rate of 3 inzlmin . At the instant that the diameter of the balloon is 2 f t, what is the late at which helium is leaking from the balloon?
[15 points] WW ad'uﬂuv ' ® 3. Is the Mean Value Theorem applicable to f(x)=—3:~ on theinterval [~3,~1] ?
x Justify your answer. If the Mean Value Theorem is applicable, find the number the value(s) of the number c guatanteed by the theorem.
[10 points] ,Hﬁa 43 @Wm [33,4] *WWMMWO. M W 4: 1(0) — A; 4—03)
Ca“ 9» 4W Q t; 4; AM (‘3,“0’
—. 3 “I
a» e I
W F981 “T 24
. 2 2 . . . d2}? .
4. Gwen x + y : xy, fmd the second denvatwe, m terms of x and y.
x ‘ ’  w ‘ [10 oiuts} P
2x+2tf$222 0631+¢J
'X dx dx inf—d);
2t} : (43~Ix)(§3 ~21) ~' (zlx)(zgz .4)
CL“ CZgvx)‘ _ t
‘ — a _.
Cay—«9"
Z (36
it)?" 2 3x? ‘3'} : axgi‘z‘é "" 33 3x( '22:)e3 (arm)
 1 ~41; — ——&————c‘l——~_
(23 70) (13‘ x); (23 7()3
{£23 2 3K ~bxt~b tit3 ~ _(e(,x1+52._’£3) w o
«w Wag«>3 " ' =
dy 5. Given the equation 2y = yzsinzx + x cosy +1, find 2x in terms of x and y, and ﬁnd the equation of the normal line to the graph of this equation at the point (— 1,0).
[9 points} 232 :: {'(Ialmiwcmoc) +Qggm‘x + %(’M8)?x +c99X L¢+xma~atamtxhih == agzmxm'x +0933 F9811IT25 3. Evaluate the following limits (in of an inﬁnite limit, determine the Sign and write
either +00 or — 00, as approppriate):
[4 points each] 3 2—
(a) m 1+x 2
«1‘ 1~x2 2
x \3 (c)1im x%:LC\~ x'irw ’5 '7‘
’Xa (d)limcscx: M ——L— : “can? KaiT“ WM ...
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